Abstract
Conditions are presented under which the solutions of asymptotically autonomous differential equations have the same asymptotic behavior as the solutions of the associated limit equations. An example displays that this does not hold in general.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blythe, S. P., Cooke, K. L., Castillo-Chavez, C.: Autonomous risk-behavior change, and non-linear incidence rate, in models of sexually transmitted diseases. Biometrics Unit Technical Report B-1048-M; Cornell University, Ithaca, NY (Preprint)
Busenberg, S., Iannelli, M.: Separable models in age-dependent population dynamics. J. Math. Biol. 22, 145–173 (1985)
Butler, G., Waltman, P.: Persistence in dynamical systems. J. Differ. Equations. 63, 255–263 (1986)
Conway, E., Hoff, D., Smoller, J.: Large time behavior of solutions of systems of nonlinear reaction-diffusion equations. SIAM J. Appl. Math. 35, 1–16 (1978)
Cushing, J. M.: A competition model for size-structured species. SIAM J. Appl. Math. 49, 838–858 (1989)
Dafermos, C. M.: An invariance principle for compact processes. J. Differ. Equations 9, 239–252 (1971)
Fiedler, B., Mallet-Paret,J.: APoincaré-Bendixson theorem for scalar reaction diffusion equations. Arch. Ration. Mech. Anal. 107, 325–345 (1989)
Grimmer, R. C.: Asymptotically almost periodic solutions of differential equations. SIAM J. Appl. Math. 17, 109–115 (1968)
Hahn, W.: Stability of Motion. Berlin Heidelberg New York: Springer 1967
Hale, J. L.: Ordinary Differential Equations, 2nd ed. Huntington, NY: Krieger 1980
Hale, J. K., Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20, 388–395 (1989)
Hirsch, M. W.: Systems of differential equations that are competitive or cooperative. IV: Structural stability in 3-dimensional systems. SIAM J. Math. Anal. 21, 1225–1234 (1990)
Hsiao, L., Su, Y., Xin, Z. P.: On the asymptotic behavior of solutions of a reacting-diffusing system: a two predator-one prey model. SIAM J. Appl. Math. 18, 647–669 (1987)
Hsu, S.-B.: Predator-mediated coexistence and extinction. Math. Biosci. 54, 231–248 (1981)
Hsu, S.-B., Cheng, K.-S., Hubbell, S. P.: Exploitative competition of microorganisms for two complementary nutrients in continuous cultures. SIAM J. Appl. Math. 41, 422–444 (1981)
Jäger, W., So, J. W.-H., Tang, B., Waltman, P.: Competition in the gradostat. J. Math. Biol. 25, 23–42 (1987)
Mallet-Paret, J., Smith, H. L.: The Poincaré-Bendixson theorem for monotone cyclic feedback systems. J. Dyn. Differ. Equations 2, 367–421 (1990)
Markus, L.: Asymptotically autonomous differential systems. In: Lefschetz, S. (ed.). Contributions to the Theory of Nonlinear Oscillations III. (Ann. Math. Stud., Vol. 36, pp. 17–29) Princeton: Princeton University Press 1956
Miller, R. K.: Almost periodic differential equations as dynamical systems with applications to the existence of a.p. solutions. J. Differ. Equations 1, 337–345 (1965)
Miller, R. K., Sell, G. R.: Volterra Integral Equations and Topological Dynamics. Mem. Am. Math. Soc. 102 (1970)
Sell, G. R.: Nonautonomous differential equations and topological dynamics. I. The basic theory. II. Limiting equations. Trans. Am. Math. Soc. 127, 241–262, 263–283 (1967)
Sell, G. R.: Topological dynamical techniques for differential and integral equations. In: Weiss, L. (ed.) Ordinary Differential Equations, pp. 287–304. Proceedings of the NRL-NCR conference 1971, New York: Academic Press 1972
Smith, H. L.: Convergent and oscillatory activation dynamics for cascades of neural nets with nearest neighbor competitive or cooperative interactions. Neural Networks 4, 41–46 (1991)
Smith, H. L., Tang, B.: Competition in the gradostat: the role of the communication rate. J. Math. Biol. 27, 139–165 (1989)
Smith, H. L., Tang, B., Waltman, P.: Competition in an n-vessel gradostat. SIAM J. Appl. Math. 51, 1451–1471 (1991)
Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Berlin Heidelberg New York: Springer 1983
Thieme, H. R.: Persistence under relaxed point-dissipativity (with applications to an endemic model). (Preprint a)
Thieme, H. R.: Asymptotically autonomous differential equations in the plane (Preprint b)
Utz, W. R., Waltman, P.: Asymptotic almost periodicity of solutions of a system of differential equations. Proc. Am. Math. Soc. 18, 597–601 (1967)
Waltman, P.: Coexistence in chemostat-like models. Rocky Mt. J. Math. 20, 777–807 (1990)
Author information
Authors and Affiliations
Additional information
Research partially supported by NSF grant DMS 9101979
Rights and permissions
About this article
Cite this article
Thieme, H.R. Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol. 30, 755–763 (1992). https://doi.org/10.1007/BF00173267
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00173267